Continuous Representations of Preferences by Means of Two Continuous Functions

Let $\precsim$ be a reflexive binary relation on a topological space $(X, \tau )$. A pair $(u,v)$ of continuous real-valued functions on $(X, \tau )$ is said to be a {\em continuous representation} of $\precsim$ if, for all $x,y \in X$, [$(x \precsim y \Leftrightarrow u(x) \leq v(y))$]. In this paper we provide a characterization of the existence of a continuous representation of this kind in the general case when neither the functions $u$ and $v$ nor the topological space $(X,\tau )$ are required to satisfy any particular assumptions. Such characterization is based on a suitable continuity assumption of the binary relation $\precsim$, called {\em weak continuity}. In this way, we generalize all the previous results on the continuous representability of interval orders, and also of total preorders, as particular cases.